Relative Cartier Divisors and Laurent Polynomial Extensions

TL;DR

This paper investigates the structure of invertible modules in ring extensions, establishing a canonical decomposition for Laurent polynomial extensions using cohomological methods, thus advancing the understanding of relative Cartier divisors.

Contribution

It introduces a new perspective on invertible modules in ring extensions by showing their contraction property and providing a canonical decomposition for Laurent polynomial extensions.

Findings

Invertible modules are contracted in ring extensions.

A canonical decomposition for Laurent polynomial extensions is established.

Cohomological methods relate invertible modules to étale cohomology.

Abstract

If $i:A\subset B$ is a commutative ring extension, we show that the group $\mathcal I(A,B)$ of invertible $A$-submodules of $B$ is contracted in the sense of Bass, with $L\mathcal I(A,B)=H^0_{et}(A,i_*\mathbb Z/\mathbb Z)$. This gives a canonical decomposition for $\mathcal I(A[t,\frac1t],B[t,\frac1t])$.